On the distribution of the number of internal equilibria in random evolutionary games

Manh Hong Duong, Hoang Minh Tran, The Anh Han

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    Abstract

    The analysis of equilibrium points is of great importance in evolutionary game theory with numerous practical ramifications in ecology, population genetics, social sciences, economics and computer science. In contrast to previous analytical approaches which primarily focus on computing the expected number of internal equilibria, in this paper we study the distribution of the number of internal equilibria in a multi-player two-strategy random evolutionary game. We derive for the first time a closed formula for the probability that the game has a
    certain number of internal equilibria, for both normal and uniform distributions of the game payoff entries. In addition, using Descartes’ rule of signs and combinatorial methods, we provide several universal upper and lower bound estimates for this probability, which are independent of the underlying payoff distribution. We also compare our analytical results with
    those obtained from extensive numerical simulations. Many results of this paper are applicable to a wider class of random polynomials that are not
    necessarily from evolutionary games.
    Original languageEnglish
    Number of pages41
    JournalJournal of Mathematical Biology
    Early online date1 Aug 2018
    DOIs
    Publication statusE-pub ahead of print - 1 Aug 2018

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